Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,21)(22,24);(10,11)(14,15)(18,20)(22,23);(1,2,4,3)(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,23)(21,24);(1,3,4,2)(9,11,13,10)(12,15,16,14)(17,20,21,18)(19,23,24,22);(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=gh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72323254, #72323255, #72323273, … |
| maximal normal subgroup of | #72323254, #72323255, #72323273, … |
| maximal quotient of | #72323688, C2×#521554, #72331623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428750);